THE MOMENT OF INERTIA 225 



Then I oz = 2a OP 2 



= Sa(OK 2 + ON 2 ) 



But la OK 2 = Sac 2 + AZ 2 



and Za ON 2 = Z0?/ 2 + Am 2 



Hence I oz = Sa(a? + t/ 2 ) + A(Z* + w 2 ) 



= Sa GP 2 + An 2 



= I GZ +An 2 ........ (3) 



where n is the distance between the axes GZ and OZ. 



It is evident from relations 1, 2, and 3, that if the moment of 

 inertia of an area about an axis passing through the centroid 

 is known, the moment of inertia about any parallel axis can be 

 found by adding the term Ad 2 where A is the area and d is the 

 perpendicular distance between the two parallel axes. 



Referring again to Fig. 57, we see that 



V 2 ) 



= Sa# 2 + Sat/ 2 

 = I GY + I GX 



Hence, if two axes are drawn at right angles to each other 

 through the centroid and in the plane of the lamina, the sum of 

 the moments of inertia about these axes will give the moment 

 of inertia of the lamina about an axis drawn perpendicular to the 

 plane of the figure and passing through the centroid. 



Now, any number of pairs of rectangular axes can be drawn 

 passing through the centroid, in the plane of the figure, but for 

 any one pair the sum of the moments of inertia is constant. 

 Thus, if the moment of inertia about one of these axes is a maxi- 

 mum, then the moment of inertia about the axis at right angles 

 to it must be a minimum. 



In dealing with questions on moments of inertia, we can there- 

 fore work with three well-defined axes. These axes are mutually 

 perpendicular and pass through the centroid. Two of these axes 

 must be drawn in the plane of the lamina and are such that the 

 moments of inertia about them are greatest and least respectively. 

 These axes are spoken of as the " Principal Axes of Inertia." 



A principal axis can also be an axis of symmetry, and if an area 

 has one axis of symmetry, this will give one principal axis, and 

 the other principal axis can be determined by drawing it through 

 the centroid and perpendicular to the axis of symmetry. 



122. Let OX and OY be two rectangular axes drawn through 

 the centroid in the plane of a given lamina (Fig. 58), and let 



P 



